Problem: Bhavik bought $3$ liters of milk and $5$ loaves of bread for a total of $\$11$. A month later, he bought $4$ liters of milk and $4$ loaves of bread at the same prices, for a total of $\$10$. How much does a liter of milk cost, and how much does a loaf of bread cost? A liter of milk costs $\$$
Let $x$ represent the cost of a liter of milk and let $y$ represent the cost of a loaf of bread. Since we have two unknowns, we need two equations to find them. Let's use the given information in order to write two equations containing $x$ and $y$. For instance, we are given that Bhavik bought $\textit{3}$ liters of milk and $\textit{5}$ loaves of bread for $\$\textit{11}$. How can we model this sentence algebraically? The total cost of milk Bhavik bought can be modeled by $3x$, and the total cost of bread he bought can be modeled by $5y$. Since these together add up to $11$, we get the following equation: $3x+5y = 11$ We are also given that a month later, Bhavik bought $\textit{4}$ liters of milk and $\textit{4}$ loaves of bread for $\$\textit{10}$. This can be expressed as: $4x+4y=10$ Now that we have a system of two equations, we can go ahead and solve it! We can now solve the system of equations by the elimination method. Let's manipulate the equations so one of the variables has the same coefficients but with opposite signs. $\begin{aligned}{-4}\cdot 3x+({-4})\cdot 5y&={-4}\cdot 11\\\\-12x-20y&=-44\end{aligned}$ $ \begin{aligned} {3}\cdot4x+{3}\cdot 4y&={3}\cdot10\\\\12x+12y&=30\end{aligned}$ Now we can eliminate $x$ : − 12 x − 20 y + 12 x + 12 y 0 − 8 y = − 44 = 30 = − 14 \begin{aligned}-12x-20y&=-44\\\\ {+}\ 12x+12y&=30\\ \hline\\ 0-8y &=-14 \end{aligned} When we solve the resulting equation, we find that $y =1.75$. Then, we can substitute this into one of the original equations and solve for $x$ to obtain $x=0.75$. Recall that $x$ denotes the cost of a liter of milk and $y$ denotes the cost of a loaf of bread. Therefore, a liter of milk costs $\$\textit{0.75}$ and a loaf of bread costs $\$\textit{1.75}$.